Bevel (modifier) each individual vertex
You mention bevel per vertex but then you proceed to illustrate beveling per edge.
Not sure what you really want to achieve, sounds like what Bevel Weights are for.
Add a Bevel Modifier to your mesh object, for ease of mental calculations set its Bevel width to $1.0$, then proceed to turn off Clamping and set the Limit Method to Weight.
Then in Edit Mode on your mesh, select desired edges and press Ctrl+E Edges &gt; Edge Bevel Weight to control the bevel width on a per-edge basis.
Edge weight acts a multiplication factor over the bevel distance, if you set bevel distance to a round number like $1.0$ it will facilitate adjustment. If now want a bevel of $0.5$ units just set the weight to $0.5$, if you want a bevel of $1$, $2$ or $3.7$ just set the weight to $1.0$, $2.0$ or $3.7$ respectively.